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Theorem suctrALTcf 29108
Description: The sucessor of a transitive class is transitive. suctrALTcf 29108, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 29109, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALTcf  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALTcf
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4661 . . . . . . . 8  |-  A  C_  suc  A
2 id 21 . . . . . . . . 9  |-  ( Tr  A  ->  Tr  A
)
3 id 21 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
4 simpl 445 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
53, 4syl 16 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 id 21 . . . . . . . . 9  |-  ( y  e.  A  ->  y  e.  A )
7 trel 4312 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
873impib 1152 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 5, 6, 8syl3an 1227 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
10 ssel2 3345 . . . . . . . 8  |-  ( ( A  C_  suc  A  /\  z  e.  A )  ->  z  e.  suc  A
)
111, 9, 10eel0321old 28900 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
12113expia 1156 . . . . . 6  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
13 id 21 . . . . . . . . 9  |-  ( y  =  A  ->  y  =  A )
14 eleq2 2499 . . . . . . . . . 10  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1514biimpac 474 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
165, 13, 15syl2an 465 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
171, 16, 10eel021old 28875 . . . . . . 7  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1817ex 425 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
19 simpr 449 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
203, 19syl 16 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
21 elsuci 4650 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2220, 21syl 16 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
23 jao 500 . . . . . . 7  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
24233imp 1148 . . . . . 6  |-  ( ( ( y  e.  A  ->  z  e.  suc  A
)  /\  ( y  =  A  ->  z  e. 
suc  A )  /\  ( y  e.  A  \/  y  =  A
) )  ->  z  e.  suc  A )
2512, 18, 22, 24eel2122old 28902 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2625ex 425 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2726alrimivv 1643 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
28 dftr2 4307 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2928biimpri 199 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
3027, 29syl 16 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
3130iin1 28737 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    C_ wss 3322   Tr wtr 4305   suc csuc 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-sn 3822  df-uni 4018  df-tr 4306  df-suc 4590
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